In a paper entitled "New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance," published in Physical Review Letters, Vol. 45, pp. 494-497 (1980), K. v. Klitzing, G. Dorda, and M. Pepper showed that the Hall resistance of a two-dimensional elctron gas, formed at the inversion layer at an interface of silicon and silicon dioxide in a metal-oxide-semiconductor field-effect transistor configuration, is quantized when this resistance is measured at liquid helium temperatures in a magnetic field of the order of 15 Tesla (150 kilogauss). By "quantized" is meant that the Hall resistance would take on certain values corresponding to Hall conductivities which were proportional to the product of the fine structure constant (approximately 1/137) and the speed of light.
In a subsequent paper entitled "Resistance Standard Using Quantization of the Hall Resistance of GaAs-Al.sub.x Ga.sub.1-x As Heterostructures," published in Applied Physics Letters, Vol. 38, pp. 550-552 (1981), D. C. Tsui and A. C. Gossard demonstrated that a two-dimensional electron gas at a heterojunction interface--specifically an interface between gallium arsenide and aluminum gallium arsenide--at sufficiently low temperatures and under sufficiently high magnetic fields perpendicular to the interface, similarly evinced quantized resistivities.
More specifically, as indicated in FIG. 1, Tsui and Gossard showed that the longitudinal or ohmic electrical resistivity (.rho..sub.xx) of the inversion layer at the heterojunction interface of a sample structure of GaAs-Al.sub.0.3 Ga.sub.0.7 As at 4.2.degree. K. exhibited minima (as a function of magnetic field) equal to substantially zero resistivity (less than 0.1 ohms per square) at magnetic fields of 4.2 and 8.4 Tesla, and that the transverse or Hall resistivity (.rho..sub.xy) exhibited stationary (quantized) values (r.sub.1, r.sub.2, r.sub.3, r.sub.4, . . . ) under these magnetic fields.
FIG. 2 illustrates an example showing how to measure these resistivity effects. As shown in FIG. 2, a source of an electrical current I is furnished by a battery of electromotive force E connected in series with a high impedance Z. The current I is supplied to the electrode contacts 11 and 12 located on opposite ends of a solid rectangular rod or bar 10 of width w and thickness t, and having a side edge 13 and an opposite side edge 14. The resistance of the impedance Z is sufficiently high that during operation the current I is essentially constant. The bar is oriented with w parallel to the y-direction and t parallel to the z-direction (perpendicular to the plane of the drawing). An inversion layer is formed at a heterojunction interface 15 (FIG. 3) between top and bottom portions 16 and 17 of the rod. The heterojunction interface 15 extends at constant z=z.sub.o all the way along a cross section of the rod 10 between the electrode contacts 11 and 12. Each of electrodes 21 and 22 of first pair of electrical probes is located in contact with the bar 10 at z=z.sub. o and at the same x coordinates, and each of electrodes 23 and 24 of a second pair of electrical probes is located in contact with the bar at z=z.sub.o and at the same x but at a distance l measured along the x-direction away from the first pair of probes 21 and 22. A uniform steady magnetic field B is applied to the bar parallel to the z-direction. As a result of the applied voltage E, a current I flows through the bar along the x-direction; as a result of the magnetic field B, a Hall effect voltage is developed across the bar 10 in the y-direction. More specifically, the voltage or potential difference V between probes 21 and 23 (or between probes 22 and 24) is measured by a voltmeter of extremely high impedance, that is, a voltmeter which draws negligible current as compared to I. Likewise, the voltage between probes 23 and 24 (or between probes 21 and 22) is also measured by a voltmeter of extremely high impedance. Accordingly, essentially no current flows in the y-direction once equilibrium is established in the bar 10 under the applied voltage E.
In accordance with the definition of the ohmic resistance R of the bar 10: EQU R=V/I (1)
On the other hand, the x-component E.sub.x of electric field in the bar 10 is equal in magnitude to V/l; and the x-component j.sub.x of the electrical current density is equal to I/wt. Accordingly, the longitudinal resistivity .rho..sub.xx, defined in this case as E.sub.x /j.sub.x, is given by: EQU .rho..sub.xx =(V/l)/(I/wt)
or EQU .rho..sub.xx /t=R(w/l)=(V/I)(w/l) (2)
Accordingly, the quantity (.rho..sub.xx /t) can be obtained from measurements of V, I, w, and l. The quantity (.rho..sub.xx /t) is called the "sheet resistivity" and thus has the same dimensions as resistance, i.e., ohms.
Moreover, as indicated above, because of the presence of the magnetic field B in the z-direction, the Hall voltage V.sub.H is developed across the width w of the bar 10, as measured across the probes 23 and 24. The corresponding Hall resistance is given by EQU R.sub.H =V.sub.H /I (3)
Accordingly, the Hall resistance R.sub.H of the bar 10 can be formed by measurements of V.sub.H and I.
On the other hand, the y-component E.sub.y of the electric field in the bar 10 is equal in magnitude to V.sub.H /w. The transverse or Hall resistivity .rho..sub.xy, defined in this case as E.sub.y /j.sub.x, is thus given by: EQU .rho..sub.xy =(V.sub.H /w)/(I/wt)
or EQU .rho..sub.xy /t=R.sub.H =V.sub.H /I (4)
Accordingly, .rho..sub.xy /t is the transverse or Hall "sheet resistivity" and also has the dimensions of ohms.
In the aforementioned paper by D. C. Tsui and A. C. Gossard, in Applied Physics Letters, Vol. 38, (.rho..sub.xx /t) and (.rho..sub.xy /t) were found to behave as indicated in FIG. 1; that is, .rho..sub.xx /t has zeros at certain values of magnetic field B, and .rho..sub.xy /t has (quantized) plateaus (r.sub.1, r.sub.2, r.sub.3, . . . ) at these values of the magnetic field B. In this sense, .rho..sub.xy /t is said to be "quantized."
More specifically, these quantized values of resistivity have been found to satisfy the relationships: EQU r.sub.1 =h/2e.sup.2 EQU r.sub.2 =h/4e.sup.2 EQU r.sub.3 =h/6e.sup.2 EQU r.sub.4 =h/8e.sup.2 ( 5)
where h is Planck constant and e is the charge on the electron. The existence of these quantized values of Hall resistivity has been shown to imply the existence of long range order in a two-dimensional electron gas. More specifically, these quantized Hall resistivities imply the existence of nonlocalized quantized states corresponding to quantum Landau levels whose wave functions extend over macroscopic distances in the inversion layer, that is, electronic states characterized by significant probability of finding an electron in the inversion layer at differing locations separated by distances typically as large as the order of millimeters. Thus far, the only important practical use of this quantized Landau level effect has been a method for accurate measurement of the value of h/e.sup.2 and hence of the fine structure constant, e.sup.2 /2.epsilon..sub.o hc=1/137, approximately, where .epsilon..sub.o is the permittivity of the vacuum and c is the speed of light. On the other hand, it would be desirable if this phenomenon of quantized Hall resistance, with its zero resistance state, could be used as a basis for switching elements and logic gates having relatively high switching speeds and low switching power-delay products.